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Theorem invghm 15130
Description: The inversion map is a group automorphism if and only if the group is abelian. (In general it is only a group homomorphism into the opposite group, but in an abelian group the opposite group coincides with the group itself.) (Contributed by Mario Carneiro, 4-May-2015.)
Hypotheses
Ref Expression
invghm.b  |-  B  =  ( Base `  G
)
invghm.m  |-  I  =  ( inv g `  G )
Assertion
Ref Expression
invghm  |-  ( G  e.  Abel  <->  I  e.  ( G  GrpHom  G ) )

Proof of Theorem invghm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 invghm.b . . 3  |-  B  =  ( Base `  G
)
2 eqid 2283 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
3 ablgrp 15094 . . 3  |-  ( G  e.  Abel  ->  G  e. 
Grp )
4 invghm.m . . . . 5  |-  I  =  ( inv g `  G )
51, 4grpinvf 14526 . . . 4  |-  ( G  e.  Grp  ->  I : B --> B )
63, 5syl 15 . . 3  |-  ( G  e.  Abel  ->  I : B --> B )
71, 2, 4ablinvadd 15111 . . . 4  |-  ( ( G  e.  Abel  /\  x  e.  B  /\  y  e.  B )  ->  (
I `  ( x
( +g  `  G ) y ) )  =  ( ( I `  x ) ( +g  `  G ) ( I `
 y ) ) )
873expb 1152 . . 3  |-  ( ( G  e.  Abel  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( x ( +g  `  G ) y ) )  =  ( ( I `  x ) ( +g  `  G
) ( I `  y ) ) )
91, 1, 2, 2, 3, 3, 6, 8isghmd 14692 . 2  |-  ( G  e.  Abel  ->  I  e.  ( G  GrpHom  G ) )
10 ghmgrp1 14685 . . 3  |-  ( I  e.  ( G  GrpHom  G )  ->  G  e.  Grp )
1110adantr 451 . . . . . . . 8  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  G  e.  Grp )
12 simprr 733 . . . . . . . 8  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  y  e.  B )
13 simprl 732 . . . . . . . 8  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  x  e.  B )
141, 2, 4grpinvadd 14544 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  y  e.  B  /\  x  e.  B )  ->  ( I `  (
y ( +g  `  G
) x ) )  =  ( ( I `
 x ) ( +g  `  G ) ( I `  y
) ) )
1511, 12, 13, 14syl3anc 1182 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( y ( +g  `  G ) x ) )  =  ( ( I `  x ) ( +g  `  G
) ( I `  y ) ) )
1615fveq2d 5529 . . . . . 6  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( I `  (
y ( +g  `  G
) x ) ) )  =  ( I `
 ( ( I `
 x ) ( +g  `  G ) ( I `  y
) ) ) )
17 simpl 443 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  I  e.  ( G  GrpHom  G ) )
181, 4grpinvcl 14527 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  ( I `  x
)  e.  B )
1911, 13, 18syl2anc 642 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  x )  e.  B
)
201, 4grpinvcl 14527 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  y  e.  B )  ->  ( I `  y
)  e.  B )
2111, 12, 20syl2anc 642 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  y )  e.  B
)
221, 2, 2ghmlin 14688 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
I `  x )  e.  B  /\  (
I `  y )  e.  B )  ->  (
I `  ( (
I `  x )
( +g  `  G ) ( I `  y
) ) )  =  ( ( I `  ( I `  x
) ) ( +g  `  G ) ( I `
 ( I `  y ) ) ) )
2317, 19, 21, 22syl3anc 1182 . . . . . 6  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( ( I `  x ) ( +g  `  G ) ( I `
 y ) ) )  =  ( ( I `  ( I `
 x ) ) ( +g  `  G
) ( I `  ( I `  y
) ) ) )
241, 4grpinvinv 14535 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  ( I `  (
I `  x )
)  =  x )
2511, 13, 24syl2anc 642 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( I `  x
) )  =  x )
261, 4grpinvinv 14535 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  y  e.  B )  ->  ( I `  (
I `  y )
)  =  y )
2711, 12, 26syl2anc 642 . . . . . . 7  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( I `  y
) )  =  y )
2825, 27oveq12d 5876 . . . . . 6  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( (
I `  ( I `  x ) ) ( +g  `  G ) ( I `  (
I `  y )
) )  =  ( x ( +g  `  G
) y ) )
2916, 23, 283eqtrd 2319 . . . . 5  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( I `  (
y ( +g  `  G
) x ) ) )  =  ( x ( +g  `  G
) y ) )
301, 2grpcl 14495 . . . . . . 7  |-  ( ( G  e.  Grp  /\  y  e.  B  /\  x  e.  B )  ->  ( y ( +g  `  G ) x )  e.  B )
3111, 12, 13, 30syl3anc 1182 . . . . . 6  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( y
( +g  `  G ) x )  e.  B
)
321, 4grpinvinv 14535 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( y ( +g  `  G ) x )  e.  B )  -> 
( I `  (
I `  ( y
( +g  `  G ) x ) ) )  =  ( y ( +g  `  G ) x ) )
3311, 31, 32syl2anc 642 . . . . 5  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( I `  ( I `  (
y ( +g  `  G
) x ) ) )  =  ( y ( +g  `  G
) x ) )
3429, 33eqtr3d 2317 . . . 4  |-  ( ( I  e.  ( G 
GrpHom  G )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( +g  `  G ) y )  =  ( y ( +g  `  G
) x ) )
3534ralrimivva 2635 . . 3  |-  ( I  e.  ( G  GrpHom  G )  ->  A. x  e.  B  A. y  e.  B  ( x
( +g  `  G ) y )  =  ( y ( +g  `  G
) x ) )
361, 2isabl2 15097 . . 3  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  A. x  e.  B  A. y  e.  B  ( x
( +g  `  G ) y )  =  ( y ( +g  `  G
) x ) ) )
3710, 35, 36sylanbrc 645 . 2  |-  ( I  e.  ( G  GrpHom  G )  ->  G  e.  Abel )
389, 37impbii 180 1  |-  ( G  e.  Abel  <->  I  e.  ( G  GrpHom  G ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   Grpcgrp 14362   inv gcminusg 14363    GrpHom cghm 14680   Abelcabel 15090
This theorem is referenced by:  gsuminv  15218  invlmhm  15799  tsmsinv  17830
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-ghm 14681  df-cmn 15091  df-abl 15092
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